# Questions tagged [circulant-matrices]

A circulant matrix is a square matrix where each row has the same elements as the previous row, cyclically rotated right by one element. It is a specific kind of a square Toeplitz matrix.

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### The eigenvectors of adding a particular rank one matrix to the circulant matrix

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...

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### Complexity of inverting and multiplying against a symmetric Toeplitz matrix with two repeated entries

I know that the computational complexity of inverting a general $n \times n$ matrix $A$ is $O(n^{2.373})$ and multiplying it against an $n \times m$ matrix is $O(n^2m)$. Moreover, I've seen that ...

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### What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ when $n$ is odd?

Suppose that $n$ is odd. The eigen values/eigenvectors of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ are successfully computed in this post.
Q. What are ...

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### Define circulant matrix using matrix-vector multiplication? [closed]

Does there exist a matrix $\mathbf{A}$ that takes any vector $\mathbf{v}\in \mathbb{R}^n$ into the circulant matrix $\mathbf{C}_{\mathbf v} = \mathbf{A}\mathbf{v} \in \mathbb{R}^{n\times n}$ ...

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### Square root of a circulant matrix block

I'm trying to show the following:
Given the following $n\times n$ symmetric circulant matrices
$$A^*=\begin{pmatrix}
1 & -\mu_a & 0 & ...&0&-\mu_a \\
-\mu_a & 1 & -\mu_a &...

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### The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$

I asked this question in MSE few days ago but there was no response.
Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\...

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### Is the random point $(C,S)$ the same as $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, with $U$ a uniform r.v.?

Suppose that $\theta_1$ and $\theta_2$ are independent and identically distributed (i.i.d.) random variables and that $\theta_j$ has probability density function (PDF) $f_j = \frac{1}{2\pi}$ ($i.e.$, ...

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### Chromatic Polynomials of Circulant Graph With Two Parameters

I have been working with the chromatic polynomials of circulant graphs of prime order $p$ with two distinct parameters, i.e.
$P_{p,i,j}(x):= P(C_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$
In ...

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### An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by
$$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...

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### Large submatrices of circulant matrices

What properties of circulant matrices are inherited by their principal submatrices? To be more specific:
Given a square (zero-one) matrix $M$ of order $n$ with all elements on its main diagonal ...

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### Fullrankness of sum of time shifts

I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...

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### Lovasz theta and circulant graphs

Let $\theta(G)$ denote Shannon zero error capacity of graph $G$ and $\vartheta(G)$ be Lovasz upper bound for $\theta(G)$.
Let $C_{2n+1}$ denote cycle graph with $2n+1$ nodes.
We know following two ...

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### Bounds for maximum determinant of circulant matrices

The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns.
An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...

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### About the properities of sum of powers of items in a polynomial

Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials:
\begin{eqnarray*}
f_2&=&a_1^2x^2+\cdots+a_{p-...

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### About the rank of a Pell equation-related matrix

I have a question about the solution of Pell-equation over a prime field.
I want prove that the matrix $M$ is of rank $\frac{p-1}{2}$, with $M=(m_{i,j})\in\left(\mathbb{Z}/(p^p-1)\mathbb{Z} \right)^{(...

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### XOR circulant matrices?

Take a function $f: Z_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z_N$. The resulting matrices are ...

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### Partial Vandermonde circulant determinant expression

Consider following partial Vandermonde type, circulant matrix
$\begin{bmatrix}
x_1 & x_2 & 0 & \dots & 0 & x_n\\
x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\
\vdots ...

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### Are certain normal matrices circulant? (Part 2)

Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $
A^TA=\begin{pmatrix}
a & b \\
b & \ddots
\end{pmatrix}$, for $b>0$.
As a user observed in the solution of Part 1 ...

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### Is a normal matrix satisfying $A^TA=...$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that
$$
A^TA=\begin{pmatrix}
a & b & \cdots & b\\
b & a & \ddots & \vdots\\
\...

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### How to determine if there exists a non-zero vector in the kernel

If you are given a $0$-$1$ circulant matrix with $n$ rows and $n$ columns, is there an efficient way of determining if there exists a non-zero $\{-1,0,1\}$-vector in its kernel?
Could this problem ...

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### Graphs with circulant distance matrices

The cycle has this property. For instance, the distance matrix for a 6-cycle is:
$A=\begin{bmatrix}
0 & 1 & 2 & 3 & 2 & 1 \\\\
1 & 0 & 1 &...

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### bounds on the entries of an inverse circulant matrix

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...

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### Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix

Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...

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### Is this general form of Lovasz theta function of circulant graphs?

Let $G$ be a cirulant graph with no loops at vertices and vertex degree $d$. Is the Lovasz theta function of this graph given by:
$\vartheta(G) = \max_{i}\frac{-N\epsilon_{i}}{-\epsilon_{i}+d-1}$?
...